Proof of addition formula in Trigonometry

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mutasimmim
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Proof of addition formula in Trigonometry

Unread post by mutasimmim » Tue Sep 09, 2014 11:48 am

Anyone has some PDF or link where I can find generalized(for any trigonometric angel) proof of the addition formula in Trigonometry? Google search yields proofs using the Euler's formula for $e^{I \theta}$. I'm looking for some other proofs.

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Phlembac Adib Hasan
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Re: Proof of addition formula in Trigonometry

Unread post by Phlembac Adib Hasan » Wed Sep 10, 2014 6:52 pm

I think it's not necessary to prove these formulae in general form. Because we know formulae like $\sin (180- \theta)=\sin \theta$, we can easily extend the classic geometrical proofs. As an example addition formula for sine is proven bellow.
1. Prove it geometrically for $A,B\ge 0$ and $A+B\leq 90$.
2. Now suppose $A$ is obtuse, $A=90+C$ and $B$ is acute. Then \[\sin(A+B)=\sin (90+B+C)=\cos(B+C)\]\[=\cos B\cos C-\sin B\sin C=\sin A\cos B+\cos A\sin B\]
3. Let both of $A$ and $B$ are obtuse and $A=90+C$ and $B=90+D$. Then\[\sin(A+B)=\sin (180+C+D)=-\sin (C+D)\]\[=-\sin C\cos D-\cos C\sin D=\cos A\sin B+\sin A\cos B\]
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mutasimmim
Posts:107
Joined:Sun Dec 12, 2010 10:46 am

Re: Proof of addition formula in Trigonometry

Unread post by mutasimmim » Wed Sep 10, 2014 7:21 pm

Thanks, Adib. And this is how they proved it in "103 Trigonometry Problems".

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